**C. Nore****, **W. Herreman. *PhD : L. Cappanera. Postdoc : J. Varela Rodriguez, H. Zaidi.*

The magnetohydrodynamic (MHD) equations describing the motion of an electrically conducting fluid couple the velocity field and the magnetic induction by the Lorentz force and the Ohm's law. MHD applications we propose to study are diverse: energy storage in **liquid metal batteries**, control of hydrodynamic instabilities by a magnetic field, generation and maintenance of a magnetic field by the turbulent motion of a conducting fluid (dynamo effect). Dynamo effect is studied in the configuration of the **von Kármán Sodium experiment**, the only one to have generated time-dependent magnetic fileds. We also study the optimization of flows to make **dynamo-capable**. These studies are conducting with different computer codes developed at LIMSI coupled with innovative models (optimization, interfaces, etc.).

#### Characterization of flows in liquid metal batteries

#### Dynamo action in the Von Kármán Sodium experiment

The Von Kármán Sodium experiment (or VKS http://perso.ens-lyon.fr/nicolas.plihon/VKS/index.php) investigates the magnetic field generation by a liquid sodium turbulent flow driven by two counter-rotating impellers consisting of disks and blades. It is the only one to have achieved dynamo regimes in 2010 showing time reversals of the magnetic induction like those of the earth's field, but for this it was necessary that the impellers be in soft iron. The role of the ferromagnetic material remains puzzling and we intend to get information on the underlying mechanism by relying on the SFEMaNS code we develop since 2002 (collab. J.-L. Guermond, TAMU, Texas). Scientific bottlenecks are first to take into account iron blades corresponding to an azimuthal variation of the magnetic permeability but also to reach the large kinetic Reynolds numbers of the flow (about 10 millions). One proposed method to remove the first bottleneck is to consider an average permeability axisymmetric and treat the azimuthal variations as a source term of the induction equation (PhD thesis of L. Cappanera). In the simplified case of kinematic dynamo where the velocity field is prescribed (chosen as the azimuthal and time-averaged turbulent experimental field), a numerical code developed at GeePs and based on the Whitney elements has led to progress in the understanding of the field of generation mechanism by ferromagnetic blades (postdoc of H. Zaidi funded by the Labex LASIPS ). To resolve the second bottleneck, a nonlinear stabilization technique will be considered to achieve large kinetic Reynolds numbers. Preliminary results in a precessing cylinder are encouraging. Once the two approaches are validated, we will couple them to compute three-dimensional numerical simulations of VKS. In a simplified model, we have also modeled the spiral vortices appearing between the blades and studied the influence of the nature of the walls on the collimation of a remnant magnetic field (post-doc of J. Varela Rodriguez funded by an InterLabex contract).

Modeling of a helical vortex (red streamlines) appearing between the impeller blades of VKS that collimates a pre-existing magnetic field: [left] the walls of the turbines are made of ferromagnetic material which intensifies the magnetic field (light blue isocontour of 5.10{-3} Tesla); [right] the walls of the turbines are made of conducting material for which intensification is lower (light blue isocontour of 1.10 {-3} Tesla). The modulus of the magnetic field is represented in a plane perpendicular to the swirl. Collaboration Interlabex, postdoc J. Varela Rodriguez.

#### Dynamo optimization

From several anti-dynamo theorems, we know that too symmetrical flows (1D parallel, 2D planar, 3D toroidal) can never act as kinematic dynamos, for whatever value of the magnetic Reynolds number Rm (comparing the inertia to the magnetic diffusivity). But this is also a very fragile situation: adding even small perturbations to these perfectly symmetrical flows can transform them into efficient dynamos. Schematically:

1D shear flows can never act as kinematic dynamos, but only need to be slightly perturbed to become dynamos.

Using a recent optimisation algorithm (A.P. Willis 2012, Phys. Rev. Lett., 109, pp 251101), we identify minimal perturbations that, added to the 1D Kolmogorov flow (an example of a 1D shear flow), transform it into a dynamo. Beneath we show the spatial structure of the perturbation velocity together with the magnetic field mode that is growing.

Spatial structure of optimal perturbation flow and magnetic field mode that is generated.

As expected, we observe that the minimal perturbation's magnitude decreases as a function of Rm, following a power law that we try to model and compare with existing analytical estimates (M.R.E Proctor 2012, J. Fluid Mech., 697, pp 504-510). This optimization method is an interesting theoretical tool, that is used here to quantitively measure the fragility of an anti-dynamo theorem.

In a collaboration with L. Chen and A. Jackson at ETH Zurich, we extend these algorithms to find optimized dynamos in spherical fluid domains.